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We determine extremal values and characterize extremal tree structures for the ratios $Ecc(T)/ecc_T(u)$, $Ecc(T)/ecc_T(v)$, $ecc_T(u)/ecc_T(v)$, and $ecc_T(u)/ecc_T(w)$ where $u,w$ are leaves of $T$ and $v$ is in the center of $T$.\n  In addition, we determine the tree structures that minimize and maximize total eccentricity among trees with a given degree sequence."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1408.5865","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-25T19:00:29Z","cross_cats_sorted":[],"title_canon_sha256":"fb1be83ebfa6542ff97a460ecbd382a9e1f06ce9847594c59f11b7d125b6ce1e","abstract_canon_sha256":"ffd8cf822b98bf69e0b943879e9002450b5c42c4cabf63cd55637e3ba19a46dc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:16:20.958584Z","signature_b64":"0jln3eMWR5E1Yv+Qw2u9S1CKKweP5Ud+z+5BmP8amt2Fq3HUmedBSZY2X+v38Bi2z9I7JBWujl6W1P/1wpxfCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8b053e8237db808414f78e5061a17c498d59adf1bc3e2886946a7bdb58ba58b8","last_reissued_at":"2026-05-18T02:16:20.957908Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:16:20.957908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Eccentricity Sums in Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Heather Smith, Hua Wang, L\\'aszl\\'o Sz\\'ekely","submitted_at":"2014-08-25T19:00:29Z","abstract_excerpt":"The eccentricity of a vertex, $ecc_T(v) = \\max_{u\\in T} d_T(v,u)$, was one of the first, distance-based, tree invariants studied. The total eccentricity of a tree, $Ecc(T)$, is the sum of eccentricities of its vertices. We determine extremal values and characterize extremal tree structures for the ratios $Ecc(T)/ecc_T(u)$, $Ecc(T)/ecc_T(v)$, $ecc_T(u)/ecc_T(v)$, and $ecc_T(u)/ecc_T(w)$ where $u,w$ are leaves of $T$ and $v$ is in the center of $T$.\n  In addition, we determine the tree structures that minimize and maximize total eccentricity among trees with a given degree sequence."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.5865","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1408.5865","created_at":"2026-05-18T02:16:20.958007+00:00"},{"alias_kind":"arxiv_version","alias_value":"1408.5865v2","created_at":"2026-05-18T02:16:20.958007+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.5865","created_at":"2026-05-18T02:16:20.958007+00:00"},{"alias_kind":"pith_short_12","alias_value":"RMCT5ARX3OAI","created_at":"2026-05-18T12:28:46.137349+00:00"},{"alias_kind":"pith_short_16","alias_value":"RMCT5ARX3OAIIFHX","created_at":"2026-05-18T12:28:46.137349+00:00"},{"alias_kind":"pith_short_8","alias_value":"RMCT5ARX","created_at":"2026-05-18T12:28:46.137349+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RMCT5ARX3OAIIFHXRZIGDIL4JG","json":"https://pith.science/pith/RMCT5ARX3OAIIFHXRZIGDIL4JG.json","graph_json":"https://pith.science/api/pith-number/RMCT5ARX3OAIIFHXRZIGDIL4JG/graph.json","events_json":"https://pith.science/api/pith-number/RMCT5ARX3OAIIFHXRZIGDIL4JG/events.json","paper":"https://pith.science/paper/RMCT5ARX"},"agent_actions":{"view_html":"https://pith.science/pith/RMCT5ARX3OAIIFHXRZIGDIL4JG","download_json":"https://pith.science/pith/RMCT5ARX3OAIIFHXRZIGDIL4JG.json","view_paper":"https://pith.science/paper/RMCT5ARX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1408.5865&json=true","fetch_graph":"https://pith.science/api/pith-number/RMCT5ARX3OAIIFHXRZIGDIL4JG/graph.json","fetch_events":"https://pith.science/api/pith-number/RMCT5ARX3OAIIFHXRZIGDIL4JG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RMCT5ARX3OAIIFHXRZIGDIL4JG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RMCT5ARX3OAIIFHXRZIGDIL4JG/action/storage_attestation","attest_author":"https://pith.science/pith/RMCT5ARX3OAIIFHXRZIGDIL4JG/action/author_attestation","sign_citation":"https://pith.science/pith/RMCT5ARX3OAIIFHXRZIGDIL4JG/action/citation_signature","submit_replication":"https://pith.science/pith/RMCT5ARX3OAIIFHXRZIGDIL4JG/action/replication_record"}},"created_at":"2026-05-18T02:16:20.958007+00:00","updated_at":"2026-05-18T02:16:20.958007+00:00"}